Describing regions in polar coordinates

Any region made up of circles, segments of circles, an annulus, or a portion of an annulus can be easily described in polar coordinates. (An annulus is a ring shaped region bounded by two concentric circles, one inside the other.) Here are a few examples:

1.

The upper half of a circle of radius 5, centered at the origin.

This region fills the plane in the coordinate region $0 \le r \le 5, 0 \le \theta \le \pi$. Note that we think of this as starting at the origin and moving along the x-axis over the range of r values, then sweeping this line through the range of $\theta$ values to form the region.

2.

One-quarter of a circle, in the first quadrant, of radius 2, centered on the origin.

This region covers $0 \le r \le 2, 0 \le \theta \le \frac{\pi}{2}$.

3.

The annular region between circles of radius 2 and 4, centered at the origin, and lying in the second and third quadrants.

Here, the radius sweeps from $2 \le r \le 4$ while the range of angles is $\frac{\pi}{2} \le \theta \le \frac{3\pi}{2}$.